Difference between revisions of "1996 USAMO Problems"
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Problems of the [[1996 USAMO | 1996]] [[USAMO]]. | Problems of the [[1996 USAMO | 1996]] [[USAMO]]. | ||
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==Problem 1== | ==Problem 1== | ||
Prove that the average of the numbers <math>n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)</math> is <math>\cot 1^{\circ}</math>. | Prove that the average of the numbers <math>n \sin n^{\circ} \; (n = 2,4,6,\ldots,180)</math> is <math>\cot 1^{\circ}</math>. | ||
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[[1996 USAMO Problems/Problem 3|Solution]] | [[1996 USAMO Problems/Problem 3|Solution]] | ||
| + | =Day 2= | ||
==Problem 4== | ==Problem 4== | ||
An <math>n</math>-term sequence <math>(x_1, x_2, \ldots, x_n)</math> in which each term is either 0 or 1 is called a ''binary sequence of length'' <math>n</math>. Let <math>a_n</math> be the number of binary sequences of length <math>n</math> containing no three consecutive terms equal to 0, 1, 0 in that order. Let <math>b_n</math> be the number of binary sequences of length <math>n</math> that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that <math>b_{n+1} = 2a_n</math> for all positive integers <math>n</math>. | An <math>n</math>-term sequence <math>(x_1, x_2, \ldots, x_n)</math> in which each term is either 0 or 1 is called a ''binary sequence of length'' <math>n</math>. Let <math>a_n</math> be the number of binary sequences of length <math>n</math> containing no three consecutive terms equal to 0, 1, 0 in that order. Let <math>b_n</math> be the number of binary sequences of length <math>n</math> that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that <math>b_{n+1} = 2a_n</math> for all positive integers <math>n</math>. | ||
Latest revision as of 07:10, 19 July 2016
Contents
Day 1
Problem 1
Prove that the average of the numbers 
is 
.
Problem 2
For any nonempty set 
of real numbers, let 
denote the sum of the elements of . Given a set 
of 
positive integers, consider the collection of all distinct sums as ranges over the nonempty subsets of . Prove that this collection of sums can be partitioned into classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.
Problem 3
Let 
be a triangle. Prove that there is a line 
(in the plane of triangle ) such that the intersection of the interior of triangle and the interior of its reflection 
in has area more than 
the area of triangle .
Day 2
Problem 4
An -term sequence 
in which each term is either 0 or 1 is called a binary sequence of length . Let 
be the number of binary sequences of length containing no three consecutive terms equal to 0, 1, 0 in that order. Let 
be the number of binary sequences of length that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that 
for all positive integers .
Problem 5
Let be a triangle, and 
an interior point such that 
, 
, 
and 
. Prove that the triangle is isosceles.
Problem 6
Determine (with proof) whether there is a subset 
of the integers with the following property: for any integer there is exactly one solution of 
with 
.
See Also
| 1996 USAMO (Problems • Resources) | ||
| Preceded by 1995 USAMO |
Followed by 1997 USAMO | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
